彈簧靜平衡平面機械臂之接頭作用力與內耗力矩分析

Abstract

串聯型機械臂已有許多應用。在過去，對機械臂之接頭作用力已進行完整的分析。然而在機械臂上有未知彈簧力時，無法分析接頭上所受之作用力，以致彈簧靜平衡機械臂之接頭作用力分析未被完整討論。對於一完美靜平衡機械臂而言，其所有接頭上之力矩需為零，由重力和彈簧力所造成的力矩可被表示為累積接頭角度的函數。在給定桿件參數下，比較具相同累積接頭角度的力矩，彈簧連接參數可被求得。但彈簧會影響所跨過的所有接頭，故需將所有接頭上之力矩方程式列出，同時考量並解出彈簧連接參數，在彈簧連接參數已知的情況下，作用力分析便可執行，求出接頭上之作用力。以四連桿例子的結果顯示，相比於沒有彈簧靜平衡之機械臂，接頭1、2和3處的接頭作用力降低了22.6%、40.1%和75.7%。另外，在彈簧靜平衡機構中，彈簧所產生之力矩一直被認為會有嚴重的內耗情況，亦即彈簧產生之力矩會相互抗衡，其被視為非用以達成靜平衡的力矩，是一種被浪費掉的力矩，如同馬達效能中，非用以驅動機構的能量，會被當作廢能來看待一樣。在前述在完美靜平衡機械臂接頭上之力矩皆需為零，其中只有重力力矩與彈簧力矩兩種力矩，故彈簧所造成的力矩可被區分為用以平衡重力力矩和非用以平衡重力力矩兩部分，彈簧內耗力矩被定義為各接頭上非用以平衡重力的彈簧力矩之潛在最大值。因為彈簧安裝參數之數量會多於力矩方程式之數量，故有部分的彈簧安裝參數非唯一解，可以被選定數值。固可藉由調整彈簧安裝參數，將彈簧內耗力矩降至最低。四連桿例子的結果顯示，通過調整彈簧連接參數，接頭 2 和 3 的內耗力矩分別降低了 28% 和 50%。此外，在靜平衡設計中，彈簧之安裝需要額外空間，並且彈簧可能會干涉到機構桿件的運動，實為應用的一大缺點。為改善此缺點，將彈簧安裝於滑動對之中，並使彈簧之拉伸或壓縮方向與滑動對之運動方向一致，此法可避免彈簧安裝時使用桿件外之空間，亦可避免彈簧在桿件運動時，與其發生干涉。為達成完美靜平衡，在平面四連桿機構中需要兩個滑動對及兩根彈簧，此兩個滑動對必須鄰接且至少有一個滑動對連接到地桿。接著可由四連桿機構進一步拓展至單自由度六連桿機構，單自由度六連桿機構分為Watt與Stephen兩種類型，前者為兩桿四連桿所組成，後者為一個四連桿與一個五連桿所組成，對於Watt六連桿機構，為避免多餘之迴圈與考量機構之強度，地桿必須為連接三接頭之桿件。此外，為降低滑動對之數量，將一滑動對安排為共用接頭。對於Stephen六連桿機構，除了前述之兩點考量，因其有五連桿迴圈，因四連桿中只有兩個旋轉接頭，故限制五連桿迴圈中亦只能有兩個非共用之旋轉接頭。此設計可使單自由度閉迴路四連桿和六連桿機構無需使用額外的空間來安裝彈簧，便可達成完美靜平衡。

Serially connected manipulators have been used in many applications. Force analysis with regard to serially connected manipulators are discussed thoroughly in the past. However, force analysis of statically balanced manipulator using springs has not been widely addressed because spring forces and motions do not share an immediate association. For the statically balanced manipulators, the torque equilibrium regarding the pre-connected joint of a typical link should be zero; and the torque contributed by gravity force and spring force can be formed as function of accumulative joint angles. Based on compatibility between torques with the same accumulative joint angle, the spring attachment parameters are constrained by given link properties. However, the spring affecting joint reaction forces of the joints crossed by the spring. It is necessary to derive the torque equilibrium equations on all joints, then attachment parameters of springs solved by these parameters simultaneously satisfying equations. Thus, spring forces and joint reaction force can be sequentially determined by a chosen set of stiffness and attached lengths of springs. The results of four-link illustrative example show that joint reaction forces are reduced by 22.6%, 40.1% and 75.7% at joint 1, 2 and 3, respectively than those without springs. And for the statically balanced manipulator, a portion of the torques caused by springs countering each other lead to an imbalance in gravitational torques and, therefore, are deemed as waste torques for springs to achieve static balance. It just like motor-torque is not used to drive the manipulator and seemed as waste energy. For the aforementioned statically balanced manipulator, the torque w.r.t the joint should be zero; in which, there are only gravitational torque and torque caused by springs. The torque contribution of spring can be classified as gravity-balancing torque and counter-torque. And the internal counter-torque is defined as the possible maximum value of these counter-torques at each joint. Because of the number of spring attachment parameters more than the number of equations in a system, the solution of parameters is not unique; some parameters can be chosen. Thus, the internal counter-toque can be minimized by adjusting the spring attachment parameters. The results of a three-DOF manipulator shows that there are 28% and 50% reductions in the internal counter-torque at joints 2 and 3, respectively, through the adjustment of spring attachment parameters. Furthermore, several static balance methods have been proposed for a spanning-spring arrangement in a statically balanced mechanism (SBM) with auxiliary elements; in which, the extra space is required for the arrangement and the auxiliary elements, and there may have interference between springs, auxiliary elements and mechanism. To avoid these disadvantages, the spring is installed in the prismatic joint; and the direction of its elongation is parallel to the moving direction of the prismatic joint. This installation method can avoid aforementioned two disadvantages, using extra space and interference with mechanism during motions. To achieve the perfect static balance, in the design, two prismatic joints are required for a closed-loop, planar four-link SBM by deriving the formulation of potential energy. And these two prismatic joints should be adjacent and perpendicular to each other. Then, the one-DOF six-link mechanism can be expended from the four-link SBM. The six-link mechanism has two types, Watt-type and Stephen-type mechanism; the former is formed by two four-link mechanism and the letter formed by one four-link mechanism and one five-link mechanism. For Watt-type mechanism, the ground-link is assumed as the ternary link to avoid redundant loop and enhance the rigidity of the mechanism. The one of prismatic joints is assumed as ternary joint to reduce the number of prismatic joints to avoid friction problem. For Stephen-type mechanism, an extra rule is applied which is there are not more than two revolute joints in the five-link mechanism except ternary joint because there are just two revolute joints in the four-link mechanism. Examples of one-DOF four-link mechanism and Stephen-type six-link mechanism show that the mechanisms reached perfectly static balance without using extra space for installing springs.